Page 221 - Contributed Paper Session (CPS) - Volume 3

CPS1999 Pranesh K. et al.
to represent data which is fuzzy and this fuzziness can be represented by the
degree of participation to a set called a membership function. Let X be a space
of points. A fuzzy set in space X is characterized by a membership
function, (), and the value of () at x representing the grade of

membership of x in A where : → [, ]. For traditional bivalent logic, the

value of membership function of crisp data can only be 0 or 1, that means,
outside the set, or within the set, respectively. However, a fuzzy set allows for
its members to have degrees between 0 and 1. Thus, it can explain natural
phenomenon more accurately. Further, conventional set theory and binary
logic have three elementary binary operations, that is, intersection (and), union
(or), and complement set (negation). The rules of binary operations were
generalized in order to fitting fuzzy data. The fuzzy logic operations truth table
is shown in Table 1. The generalized form of the operators works well for the
fuzzy and for the bivalent data as well.

Table 1: The Generalized form of operations; Truth table
min(, )
max(, )
1 −

2. Methodology
The classical linear regression has crisp coefficients and is bounded by
some strict assumptions about the given data, that is, the unobserved error
terms are mutually independent and identically distributed. However, if the
data set is too small in size, or, if there is difficulty in verifying that the errors
are normally distributed, or, if there is vagueness in the relationship between
the dependent and independent variables, or, if there is ambiguity associated
with the events, it is well known that the classical linear regression may fail to
work satisfactorily. In such cases, alternatively, fuzzy linear regression may be
more useful. Fuzzy linear regression (FLR) was first introduced by Tanaka
(1982) and then further developed in Tanaka (1987). The FLR model includes
a fuzzy output and non-fuzzy input variables and fuzzy coefficients. In this
paper, however, our focus is on the type of fuzzy regression model considered
by Tanaka (1987). The basic model assumes a fuzzy linear functional form

̃ = + + ⋯ + , (1)
̃
̃
̃
1 1
0

where = [ , , … , ] is a vector of input variables, = [ , , … , ] is a
̃
̃
̃
̃

0
1

1
2

vector of fuzzy coefficients presented in the form of symmetric triangular fuzzy
data denoted by = ( , ) with its membership function described as
̃

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